How to extend curve on another curve
This should work for the latest release of 2.80 beta.

For one thing, your original curve is not periodic--this means that if you try to link them end-to-end, the ends aren't going to match up. I would create a new curly object so that it will connect end-to-end with itself without having to perform additional rotations.

After this is done, make sure that the origins for the curly and and path are centered to geometric means and that their rotations and scale transformations have been applied.

Apply an array modifier to the curly object. choose "Fit Curve" as the "Fit Type:" and select the path object as the "Curve:".

Next apply the curve modifier and select the same path as the curve you are fitting to.

Using the X Axis as the "Deformation Axis:" tends to work best for the curve modifier but experiment with other axes if the curly's aren't arranged end-to-end. Here's something you might not have tried that I needed to do (although probably not since your path is closed): You can move and rotate the curly object to fine-tune its position on the path. I had to do this when I made these balusters follow a path that was curved like the banister: 



Turn on "Merge" in the array modifier. Keeping the "Relative Offset:" value at "1.00" should string the curly end-to-end but you can adjust it if you have to. If you are close but the ends don't meet up, convert the whole thing to a mesh under the Object -> Convert To menu and move the vertices manually.

Comment if you have additional problems or questions.

Don't overthink "periodic", we are just talking about a repeating pattern. Here is a piece of the sin(x) wave function that repeats, for example:


Any piece of this function that is this long will repeat end-to-end.

HOW TO CREATE PERIODIC CURVE THE EASY WAY:

1) Create a Path object (Nurbs path or Bezier, doesn't matter, just make sure it is straight). Rotate the object so that it is oriented parallel to the z axis. Offset it from the z axis by 1 unit and move the object's origin to the world origin, as shown:



I decided to scale the path in the z-direction by half (personal choice):



Now add a SimpleDeform modifier. Make sure the option is set to Twist, the axis is set to X and the angle is set to 360 degrees (which is a complete revolution). Now your path is periodic.

 

Top view:



yet another view:



Now you can give the curve some bevel depth and convert it to a mesh (Object -> Convert To).